The question asked was about yeshiva bochurim who will be going to their camps on Monday, the day after Tisha B'Av, and need to do laundry Sunday night to prepare for their trip away.

To point out - the custom is that during the Nine Days we do not do laundry (caveat that Sephardim do laundry except for the week of Tisha B'Av itself and there are possibly some other customs as well). This has been extended to the night after Tisha B'Av to commemorate that the main burning of the midash actually happened on 10 Av so we refrain from laundry and other restrictions until noon of 10 Av.

So the Rosh Yeshiva asked the Rav was about these young men who cannot wait until the 10th because they will be going away and need the laundry den before that? And also, what about for people who will be getting married on Monday, can they do laundry on Sunday night in preparation for the wedding?

Rav Kook answered with an explanation he heard from his rebbi, Rav Elyashiv.

Rav Kook said that the bride and groom themselves (not guests or family members) can start doing the needed laundry on Sunday night, while other family members can start doing their laundry Monday morning without waiting for mid-day. People who will be traveling on Monday can do laundry on Monday morning before mid-day. At the end of the day, the custom to not do laundry is just a custom.

I am not quite sure what that last comment means. Customs are considered pretty important, especially customs brought in the actual Shulchan Aruch and commentaries.. dismissing it as "just a custom" seems out of character, but it gives me leeway on lots of things! 

Regarding the campers laundry question, Rav Kook said that the comment about someone traveling on 10 Av was regarding someone traveling to perform a mitzva. It is not clear if one is traveling not for a mitzva but for personal reasons and benefits if this would be allowed. Rav Kook concludes though that if any aspect of a mitzva is involved, one can be lenient and do laundry on Sunday night. Yeshva bochurim going on Monday, 10 Av, to camp under the auspices of the yeshiva, is surely involving a mitzva in some way as it will include chizzuk, or inspiration and the boys strengthening themselves religiously, so they can surely do laundry on Sunday night. Better, if possible, to wait until the morning but if that is not possible, laundry can be done Sunday night. 

source: Behadrei